Partial Sums I

Suppose that $X_1, \dots, X_n$ are all IID random variables with mean $\mu$ and variance $\sigma^2$. Define $S_j = X_1 + \dots + X_j$, with $S_0 = 0$. If possible, compute $\mathbb{E}\left[ S_k \mid S_n \right]$, where $0 \leq k \leq n$. Report the answer when $X_1,\dots, X_n \sim \exponential{1}$ IID, $k = 4, n = 10,$ and $S_{10} = 15$.